Name for the real numbers between $0$ and $1$ I see this class of numbers all the time, so I was wondering if there was a special name for it.
How to refer to a number $n$ in $\Bbb R$, such that $0<n<1$? 
 A: The set of all such numbers is $\{x \in \Bbb{R} \mid 0<x<1\}$, which is also more simply denoted $(0, 1)$, and occasionally, as: $\;]0, 1[\;$. 
As Andrew pointed out, it is often referred to as "the open unit interval". 
(One could say that the unit interval is to the real numbers what the unit circle is to the complex numbers, so to speak.)
One reason the values in the unit interval come up a lot is that the open unit interval is often used as an exemplar/representative of the real numbers, what can be said of the open unit interval can often be said of the set of real numbers, and vice versa. In terms of set theory, the cardinality of the unit interval of real numbers is equal to the cardinality of the real numbers; there exist bijections between the unit interval and the set of real numbers. And the range of probability is usually encompassed by the (closed) unit interval. So yes, it comes up a lot... 
If you are asking whether there is some common name for referring to some particular element $x \in \Bbb{R}$, such that $0<x<1$ (i.e. if you are asking for the name an element of the set $(0,1)$ ...):
Then I would simply refer to $x$ as "an element in the open unit interval (of $\Bbb{R}$)" and you could simply denote such an element by writing $x\in (0,1)\subset \mathbb{R}$.
